The CSA could win very easily in many ways. Death of Lincoln, worse riots in New York, the union pisses off Kentucky into not allowing troops in their territory or even outright joining the Confederacy, European intervention. You don't have to have them win the battles of OTL. As an exercise of answering OPs question I think it's fair enough to just assume they've won and figure out the multiple directions they can go from there without discounting the less-than-absurd.
It's fair to assume they'd have expansionist policies, particularly starting with a Pacific coast through Mexico and then Cuba. Depending on how that would go they'd either move on to Latin America or stop right there. The Union, if it follows the policies of OTL, may also continue it's own pursuit of banana republics. CSA, if possible, would make NIMBY it's rhetorical policy, but it just may lead to a scramble. Both countries would have enormous incentives for a transoceanic canal at some point.
In a different alternate timeline things could go much worse for sure. The CSA wins at massive cost, Texas soon secedes, slave rebellions break out and are brutally put down, US intervenes and annexes territories without total reacquisition. At that point if the CSA is from Virginia to Florida to Louisiana you can forget about further expansionism. Depending on their treatment of the black folks we can find them in pariah situation if they even exist long enough to witness modern globalism.